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Subdyadic square functions and applications to weighted harmonic analysis (1510.01897v2)
Published 7 Oct 2015 in math.CA
Abstract: Through the study of novel variants of the classical Littlewood-Paley-Stein $g$-functions, we obtain pointwise estimates for broad classes of highly-singular Fourier multipliers on $\mathbb{R}d$ satisfying regularity hypotheses adapted to fine (subdyadic) scales. In particular, this allows us to efficiently bound such multipliers by geometrically-defined maximal operators via general weighted $L2$ inequalities, in the spirit of a well-known conjecture of Stein. Our framework applies to solution operators for dispersive PDE, such as the time-dependent free Schr\"odinger equation, and other highly oscillatory convolution operators that fall well beyond the scope of the Calder\'on-Zygmund theory.